Abstract
The simplest topologically ordered phase in is the deconfined phase of gauge theory (realized in the toric code, for example). This phase permits a duality that exchanges electric and magnetic excitations (“” and “” particles). The phase transition where one of these particles condenses, while the other remains gapped, has 3D Ising exponents. But the transition out of the deconfined phase when self-duality symmetry is preserved is more mysterious. It has so far been unclear whether this transition is continuous, but if continuous, it may be the simplest critical point for which a useful continuum Lagrangian is still lacking. These questions are relevant to soft matter, too, since the gauge theory also describes classical membranes in 3D. Here, we study the self-dual transition with Monte Carlo simulations of the gauge-Higgs model on cubic lattices of linear size . Our results indicate a continuous transition, for example via a striking parameter-free scaling collapse. We use duality symmetry to distinguish the leading duality-odd and duality-even scaling operators and . We explain why standard techniques for locating the critical point are ineffective, and we develop an alternative using “renormalization group trajectories” of cumulants. We check that two- and three-point functions are scale invariant, with scaling dimensions and (autocorrelations in the Monte Carlo dynamics also yield a dynamical exponent ). Separately, we propose a general picture for emergent 1-form symmetries, in terms of “patching” of membranes or world surfaces. We relate this to the percolation of anyon worldlines in spacetime. The latter yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.
- Received 28 March 2021
- Revised 14 June 2021
- Accepted 20 July 2021
DOI:https://doi.org/10.1103/PhysRevX.11.041008
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
A phase transition is the point where, as conditions are varied, a complex system switches from one pattern of organization to another. The theory of such transitions lays the cornerstone of our understanding of complex systems. The simplest phase transition that is not yet understood may be the exit from the “deconfined” phase in self-dual Ising gauge theory. Using computer simulations and theoretical arguments, we explore what happens at this unusual phase transition, which existing theory tells us very little about.
One way to think of the Ising gauge model is as a dense, complex system of randomly shaped membranes that fluctuate, break, and reform according to the rules of statistical mechanics. Such membranes display various phases, the most unusual of which is known as the sponge phase (in the terminology of real lipid membranes) or the deconfined phase (in the terminology of gauge theory). Here, sheets of membrane connect in an infinite network of complex topology.
The mystery is what happens when we vary conditions so as to exit the sponge or deconfined phase, while preserving a key symmetry of the problem known as self-duality. We find that this transition is “critical”: Instead of suddenly jumping from the sponge phase to a simpler phase, the system organizes itself in a fundamentally new state at the transition point. We obtain the indices characterizing the critical state and provide new ways of thinking about the emergent properties of the membrane system, such as its symmetries.
For the future, there is much to explore about this unique critical state, from its relation to quantum field theory to its realization in experiments, either in the quantum or the classical domain.
